Forming the equation of a line, given the coordinates of a point on it, and its slope

Given the coordinates of a point (x₁, y₁) that lies on a particular line, and the slope of the line 'm', its equation can be obtained by the following

(y - y₁) = m(x - x₁)

This is known as the point-slope formula, or the point-slope form of the equation of a line. On substituting the values of x₁, y₁ and 'm' in the above equation, the equation of a line, containing the point (x₁, y₁) and having a slope of 'm', is derived.


For example, give a point (1, 10) and the slope m = 2 for a line, its equation is derived as follows:

(y - y₁) = m(x - x₁)

(y - 10) = 2(x - 1)

The values of x₁ , y₁ are obtained from the point (1, 10), where x₁ = 1 and y₁ = 10, while 'm' is the slope of the line, given 2.

Simplifying the above equation, and writing it in slope-intercept form,

(y - 10) = 2(x - 1)

y - 10 = 2x - 2

y = 2x - 2 + 10

y = 2x + 8,

which is the equation of the line containing the point (1, 10) and having a slope of 2.

Writing the equation in standard form,

(y - 10) = 2(x - 1)

y - 10 = 2x - 2

-2x + y = -2 + 10

2x - y = -8

which is the same equation in standard form.

Graph of  y = 2x + 8 with point (1, 10) on it: